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Nonlinear Ordinary Differential Equations in Fluid Dynamics
American Journal of Physics
  • John D. Ramshaw, Portland State University
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Publication Date
  • Nonlinear differential equations,
  • Fluid dynamics,
  • Partial differential equations

The equivalence between nonlinear ordinary differential equations (ODEs) and linear partial differential equations (PDEs) was recently revisited by Smith, who used the equivalence to transform the ODEs of Newtonian dynamics into equivalent PDEs, from which analytical solutions to several simple dynamical problems were derived. We show how this equivalence can be used to derive a variety of exact solutions to the PDEs describing advection in fluid dynamics in terms of solutions to the equivalent ODEs for the trajectories of Lagrangian fluid particles. The PDEs that we consider describe the time evolution of non-diffusive scalars, conserved densities, and Lagrangian surfaces advected by an arbitrary compressible fluid velocity field u(x, t). By virtue of their arbitrary initial conditions, the analytical solutions are asymmetric and three-dimensional even when the velocity field is one-dimensional or symmetrical. Such solutions are useful for verifying multidimensional numerical algorithms and computer codes for simulating advection and interfacial dynamics in fluids. Illustrative examples are discussed.


Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Association of Physics Teachers. The following article appeared in American Journal Of Physics, 79(12), 1255-1260; and may be found at:

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Ramshaw, J. D. (2011). Nonlinear ordinary differential equations in fluid dynamics. American Journal Of Physics, 79(12), 1255-1260.